L(s) = 1 | + (0.5 − 0.866i)2-s + (0.5 + 0.866i)3-s + (−0.499 − 0.866i)4-s + (−1.30 + 2.26i)5-s + 0.999·6-s + (1.77 − 1.95i)7-s − 0.999·8-s + (−0.499 + 0.866i)9-s + (1.30 + 2.26i)10-s + (1.77 + 3.08i)11-s + (0.499 − 0.866i)12-s + 13-s + (−0.806 − 2.51i)14-s − 2.61·15-s + (−0.5 + 0.866i)16-s + (3.97 + 6.88i)17-s + ⋯ |
L(s) = 1 | + (0.353 − 0.612i)2-s + (0.288 + 0.499i)3-s + (−0.249 − 0.433i)4-s + (−0.584 + 1.01i)5-s + 0.408·6-s + (0.672 − 0.740i)7-s − 0.353·8-s + (−0.166 + 0.288i)9-s + (0.413 + 0.715i)10-s + (0.536 + 0.929i)11-s + (0.144 − 0.249i)12-s + 0.277·13-s + (−0.215 − 0.673i)14-s − 0.674·15-s + (−0.125 + 0.216i)16-s + (0.964 + 1.67i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.931 - 0.364i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.931 - 0.364i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.79746 + 0.339334i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.79746 + 0.339334i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.5 + 0.866i)T \) |
| 3 | \( 1 + (-0.5 - 0.866i)T \) |
| 7 | \( 1 + (-1.77 + 1.95i)T \) |
| 13 | \( 1 - T \) |
good | 5 | \( 1 + (1.30 - 2.26i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (-1.77 - 3.08i)T + (-5.5 + 9.52i)T^{2} \) |
| 17 | \( 1 + (-3.97 - 6.88i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-1.08 + 1.88i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-1.27 + 2.21i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + 3T + 29T^{2} \) |
| 31 | \( 1 + (-0.414 - 0.717i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (3.89 - 6.74i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 - 11.7T + 41T^{2} \) |
| 43 | \( 1 + 0.773T + 43T^{2} \) |
| 47 | \( 1 + (-1.47 + 2.55i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (6.28 + 10.8i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (2.39 + 4.14i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (3.36 - 5.82i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (7.25 + 12.5i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 7.82T + 71T^{2} \) |
| 73 | \( 1 + (5.27 + 9.14i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-1.19 + 2.07i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 15.5T + 83T^{2} \) |
| 89 | \( 1 + (2.10 - 3.65i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 - 10.9T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.71429082063883570979683203792, −10.37084494221813223005474303480, −9.324480482809467995739554270722, −8.137871819110295681236988840341, −7.33934610710358968093364410885, −6.27659396011283165035589279879, −4.84328553146004376769497771705, −3.96285091442084637246653561786, −3.23968560466899010679009961817, −1.68133929066394928891195262419,
1.07250534433156500902254939539, 2.95545909669024976087398218715, 4.19075974935540139489128494179, 5.31199508667136995157043206532, 5.95009963967469842967297547834, 7.43275636453549136814637379778, 7.912151073363825105650732267670, 8.935138428452857564993020307798, 9.249552254833200396446797703181, 11.11528265998350990427414807716